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I have only ever seen the Schrodinger equation for the hydrogen atom written out in a form like this: $$ -\frac{\hbar^2}{2\mu}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial\psi}{\partial\theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2\psi}{\partial \phi^2}\right]-\frac{Ze^2}{4\pi\epsilon_0 r}\psi=E\psi $$

I'm still learning the necessary skills to solve PDEs, let alone get to the point of solving this problem, but I wanted to know if someone could show me what this differential equation would look like in a matrix notation or as a system of differential equations.

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    $\begingroup$ That looks like the Schroedinger equation for the hydrogen atom in radial coordinates. The solution for it should be in every introductory QM textbook. What kind of "matrix notation" are you looking for? Heisenberg's matrix mechanics? Strictly speaking that's only a good way of looking at problems with a discrete spectrum (which this is not). I think you should try to learn modern methods of solving this problem rather than trying to fall back on initial attempts to formulate quantum mechanics. As for PDEs, it would be better to take a math class first. $\endgroup$ – CuriousOne Oct 14 '14 at 8:27
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    $\begingroup$ @CuriousOne: The hydrogen system most certainly has a discrete spectrum, good sir/madam. $\endgroup$ – DanielSank Oct 14 '14 at 12:07
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If you assume separability of the wave function, i.e., $\psi(\mathbf x)=u(x)v(y)w(z)$, you can solve the individual components separately: \begin{align} -\frac{\hbar^2}{2\mu}\frac{d^2u(x)}{dx^2}+V_1(x)u(x)&=E_1u(x)\\ -\frac{\hbar^2}{2\mu}\frac{d^2v(y)}{dy^2}+V_2(y)v(y)&=E_2v(y)\tag{1}\\ -\frac{\hbar^2}{2\mu}\frac{d^2w(z)}{dz^2}+V_3(z)w(z)&=E_3w(z) \end{align} with the further constraint that $$ E_1+E_2+E_3=E $$

We can express (1) as the matrix differential equation, $$ \mathbf u''=A\mathbf u,\tag{2} $$ in which case $A$ is clearly diagonal and $\mathbf u=(u(x),\,v(y),\,w(z))^T$. In the case that the wave-function is not separable, then this method is not appropriate as you'd have a single scalar equation.

For your case of the spherical wave function, you can solve the radial component and the angular component separately, $\psi(\mathbf r)=R(r)Y(\theta,\phi)$ with $Y(\theta,\phi)$ the spherical harmonics, as \begin{align} \frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR(r)}{dr}\right)&=\lambda \\ \frac1Y\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial Y}{\partial\theta}\right)+\frac1Y\frac1{\sin^2\theta}\frac{\partial^2Y}{\partial\phi^2}&=-\lambda \end{align} where $\lambda$ is a parameter to be discovered. This is the typical method of solving this particular problem in quantum mechanics textbooks.

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